Herzog Competition - 2022

The traditional Herzog competition of year 2022 took place on November 19. Seventeen MSU undergraduate students had three hours to solve six problems. The winners are

Here is the list of problems

Question 1

Find all natural $a$ and $b$ such that neither of these two numbers is a multiple of the other number and

Question 2

Fibonacci numbers $F_n$ are defined as $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n\ge 3$. Let $\mathrm{S}(r)$ denote the number of possibilities to write $F_r$ as a sum of Fibonacci numbers with distinct numbers (say, $F_r=F_r=F_{r-1}+F_{r-2}=\cdots$). Find $\mathrm{S}(2022)$.

Question 3

Let $f:\mathbb{R}\to \mathbb{R}$ be an everywhere differentiable function defined for all real $x$ and let

for any $x\in \mathbb{R}$. Prove that $\mathrm{f}(x)$ is a linear function, that is, $\mathrm{f}(x)=ax+b$ for some $a,b\in \mathbb{R}$.

Question 4

Define functions $Q_n(x)$ recurrently, $Q_0(x)=1$, $Q_1(x)=x$ and

Prove that $Q_n(x)$ is a polynomial for all natural $n$.

Question 5

Let a triangle-shaped solid angle be bounded by three edges parallel to three unit vectors $i_1$, $i_2$, and $i_3$ in ${\mathbb{R}}^3$ and by three sides lying in the planes spanned by $(i_1,i_2)$, $(i_1,i_3)$ and $(i_2,i_3)$. Define the “mean line” of such a solid angle as the line passing through the summit of the angle in the direction $i_1+i_2+i_3$.

Prove that the mean lines of all four solid angles of a tetrahedron pass through the same point inside the tetrahedron if and only if all pairwise products of lengths of opposite sides of the tetrahedron are equal to the same number.

Question 6

Find all positive integer solutions $x,y,z$ to the equation